A. R. Mekala et al. Int. Journal of Engineering Research and Application Vol. 3, Issue 5, Sep-Oct 2013, pp.317-323
RESEARCH ARTICLE
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OPEN ACCESS
Energy-Efficient Power Allocation And Performance Estimation For MIMO-MRC Systems Ananda Reddy Mekala1, Penchalaiahyadla2, Irala Suneetha3, Anitha Mekala4 1,2,3,4
(Department of Electronics and CommunicatoinAnnamacharya Institute of Technology and Sciences, Tirupati)
ABSTRACT Adaptive power allocation problem where it minimizes the energy-per-good bit (EPG) of a system employing a multiple-input multiple-output maximum ratio combining (MIMO-MRC) scheme is formulated. Closed-form results are obtained for the optimum transmit power and minimum EPG as a function of the number of antennas employed and the quality of the channel. The cumulative distribution function (CDF) of the minimum EPG in closed form is obtained to assess the performance of the solution in a statistically varying channel. The energyefficiency trade-off between enhanced diversity and the increased circuit power consumption of multiple antennas is explored. In particular, EPG CDF in a numerical example is used to find the most energy-efficient number of antennas for a given probability of outage. Both Rayleigh and Rician MIMO fading channels are considered. Keywords-Energy minimization, MIMO, beamforming, MIMO MRC, fading channels, adaptive power allocation. Introduction. possible in this work by simplifications in the EPG model (described in section II-C). The contributions I. INTRODUCTION of this letter are: Adaptive power allocation or more generally 1) Problem formulation: the objective function, adaptive modulation, where the transmitter adapts its energy-per good bit(EPG), is extended to a transmit power, coding, modulation or any MIMO-MRC system combination thereof in response to the channel fading, 2) Closed-form solutions for energy-efficient transmit is a technique that can provide substantial gains in power, spectral efficiency and minimum EPG spectral efficiency and reliability. Recently adaptive 3) Statistical performance analysis: CDF of the modulation has been used to provide substantial gains minimum EPG is characterized for any fading in energy efficiency especially in cases where the distribution energy consumed by the communication circuitry is 4) Numerical example: demonstrates application of not negligible. It is well known that multiple antennas closed form results by characterizing the best at the transmitter and receiver or multiple-input MIMO configuration as a function of the ratio of multiple output (MIMO) can provide dramatic circuit power to transmit power costs. improvements in spectral efficiency and reliability without requiring an increase in bandwidth or power. The paper is organized as follows. In section However, using multiple radio chains incurs a higher II, the EPG objective function for the MIMO-MRC circuit power consumption. In this paper the trade-off system is Introduced and the energy minimization between the diversity advantage of multiple antennas problem is formulated. In section III, we derive and the higher circuit power consumption by focusing closed-form results for the optimum transmit power on energy efficiency of the wireless link is studied. and minimum EPG. In section IV, we characterize the Most works in the area of energy-efficient statistical performance of the minimum EPG by communications either involve solving a convex deriving its CDF. In section V, we present a numerical optimization problem or evaluate the solution by an example where we use the results derived to evaluate exhaustive search. Some of this work has also been the most energy-efficient MIMO-MRC configuration extended to the MIMO scenario, where (in terms of the number of antennas) under both Alamouti space-time block code (STBC)is Rayleigh and Rician fading conditions. In this paper, used. In contrast to all of these works, closed form the terms rate, bits-per-symbol and spectral efficiency solutions are provided. Also unlike, perfect channel are considered equivalent and are used interchangeably. state information (CSI) at the transmitter and hence A. Notation focus on beamforming is assumed . In contrast to Vectors and matrices are denoted with boldface in which relied on simulation to analyze performance, a lower andupper case, respectively. The operations ()T complete statistical characterization of the solution and ()†representtranspose and conjugate transpose (closed-form CDF of minimum EPG) is made www.ijera.com
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A. R. Mekala et al. Int. Journal of Engineering Research and Application Vol. 3, Issue 5, Sep-Oct 2013, pp.317-323 operations, respectively.The symbol ∇ represents the gradient operation.
II.
PROBLEM FORMULATION
A. System preliminaries Bits from a packet of length L bits are coded and modulated, where coded bits are mapped to a constellation symbol, x, at an average power GP, where P is the component of transmit power that will be optimized and G is defined as the constant transmit power that is needed to overcome deleterious effects such as path loss, implementation loss and thermal noise, but also takes into account the benefits of coding gain (or equivalently SNR gap) and antenna gain. The constant transmit power G is defined as
Where ML is the link margin to account for implementation imperfections in the system, λ is the wavelength at the carrier frequency, d is the distance between the transmitter and receiver and κ is the pathloss exponent based on a reference distance of 1 meter. We also scale G by the variance of the additive thermal noise observed at the receiver. This noise is modeled as additive white Gaussian noise (AWGN) with variance , where is the two-sided noise power spectral density (PSD), Nf is the noise figure of the receiver and B is the bandwidth of the system. Assuming a fixed operating bit error rate (BER), BER, we can compute the SNR gap approximation (BER) for a given coded modulation scheme. As a simple example, an approximate SNR gap (from Shannon limit) for uncoded QAM is . It is assumed a packet based system with ARQ in a long-term static fading channel. We compute packet error rate (PER) using the formula given by PER = 1 −(1 −BER)L, which is an approximation. We assume that the receiver and transmitter have perfect channel state information (CSI). B. Channel model In a MIMO system with receive antennas and transmit antennas, the received signal , after some simplifications, is a linear transformation of transmit signal plus an additive noise where is the MIMO channel matrix and whas i.i.d. complex Gaussian elements with zeromean and unitvariance. We consider scalar beamforming schemes in thiswork, i.e. s is a function of only one information symbol. Thetransmit signal depends on unit-norm transmit beam forming vector v and scalar (information symbol from unitenergyconstellation A) as . The received www.ijera.com
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signal y islinearly combined by receive beamforming vector u† to yield the scalar decision statistic . Assuming coherent detection at the receiver, the resulting channel model is given by Where
is zero-mean unit-variance AWGN and , effectively is the channel gain-to-noise ratio (CNR). By applying the Cauchy-Schwartz inequality, we have
Thus, the maximum CNR is achieved by setting v to the eigen vector corresponding to the maximum eigenvalue of H†H, max(H†H), and u = Hv. For a Single Input Multiple Output (SIMO) configuration (where and ),this scheme is called Maximum Ratio Combining (MRC), fora Multiple Input Single Output (MISO) configuration (where and ), this scheme is called Maximum RatioTransmission (MRT) and for a general MIMO configuration,this scheme is called transmit-beamforming (TB) or MIMO-MRC, which is illustrated in Fig. 1.
Fig. 1. The architechture of nR × nT MIMO-MRC configuration. C. Objective function: energy-per-goodbit (EPG) In this paper, we formulate EPG as a function of the variable P, which corresponds to transmit power without the constant scale factor G. Due to normalizations in G(see (1)), effectively the signal to noise ratio (SNR) at the receiver is γP. The bits-per-symbol or spectral efficiency isgiven by . The EPG for a single carrier MIMO-MRC system is defined for γ >0 as
Where and are non-negative constants which are interpreted as transmit and circuit EPG respectively for unit transmit power and unit spectral efficiency, and are given by and , respectively. Parametersin the model include, , where OBO is the outputbackoff from saturation power level of the power amplifier(PA) and is the maximum PA efficiency (or drainefficiency), is the average power consumption in a singletransmit or receiver chain (the rest of the 318 | P a g e
A. R. Mekala et al. Int. Journal of Engineering Research and Application Vol. 3, Issue 5, Sep-Oct 2013, pp.317-323 transmit and receiveelectronics excluding the PA). For a MIMO system, is computed as
Where and are the power consumed in a singletransmit chain and receive chain, Pco is the (common) power consumed independent of the number of antennas and the total number of antennas in the system, .In contrast to the EPG model , the above model assumes a constant PA backoff. This assumption can be a practical one where the back-off is a function of the maximum supported constellation. Orthogonal frequency division multiplexing (OFDM) is another example where a constant back-off may be employed in practice. It is to be noted that the results presented in this paper are directly applicable for OFDM systems under a frequency-flat fading channel model. A constant backoff also may be prescribed for coded modulation systems, where the code rate is adapted and the constellation size is fixed. The constant PA back-off assumption facilitates the derivation of a closed-form solution for the distribution function of the optimum EPG. D. Optimization problem For a given MIMO configuration and channel realization, H, the MIMO-MRC scheme results in a CNR . The goal is then to find the power allocation, , that results in the minimum EPG, , subject to certain quality of service (QoS) constraints, that is,
where is the optimum feasible solution over the convex set X defining the set of feasible power allocations.
III.
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valued, positive, differentiable, concave function both defined on an open convex set ⊂ , then on X is pseudoconvex. Proposition1:
is pseudo convex on .
Proof:
is of the form f(P)g(P), where is affine and non-negative since and are non-negative, and is concave in P and over X. Note that is only defined for > 0. By theorem 2, is pseudoconvex on X. B. Optimum power allocation and minimum EPG Using proposition 1 and theorem 1, the optimum power allocation, is the solution to . Assuming that > 0 (since is not defined otherwise) and after performing some simple manipulations is the solution to where is defined as the ratio of circuit to transmit EPG for unit rate and power, i.e. . Using the Lambert W function and (11) from the appendix, the optimum power thatminimizes the EPG as a function of is
In practical systems, the spectral efficiency may be restricted to a finite set of values as dictated by allowed constellation sizes and code rates. In these cases, a nearest neighbor rounding of the solution can be employed. The expression for the minimum EPG is obtained by plugging the solutions in (4) and applying (12) from the appendix
ENERGY–EFFICIENCT POWER ALLOCATION
A. Pseudoconvexity of objective function A definition is provided for a pseudo-convex function and then present some theorems which will be useful for our purposes. Definition 1: Let X be a nonempty open convex set in Rn and let h : X → R be differentiable on X. Then h is pseudoconvex if for each we have ∇ . Theorem 1: Let h be a differentiable pseudo convex function over X ⊂ Rn which is an open convex set and suppose that ∇h(x ) = 0 for some x X. Then x is a global minimum of h over X. Proof: From the definition it is clear that for all . Although pseudoconvex functions do not have to be convex, they possess an important attribute of convex functions, Theorem 2: If f(x) is a real-valued, non-negative, differentiable, convex function and, g(x) is a realwww.ijera.com
From the expression above, we can immediately conclude that the minimum EPG is positive for all > 0. Also for any power allocation P, it is clear that the EPG in (4) is strictly decreasing in . Thus, intuitively, we expect the MIMO-MRC scheme (which maximizes over all possible beamforming schemes) to be the EPG optimal beamforming scheme. In fact, it can be shown that the minimum EPG is a decreasing function of .
IV.
PERFORMANCE ESTIMATION
In the previous section it was shown that the optimum power allocation and consequently the minimum EPG that results, is a function of the channel fade power . If we treat as a random variable with CDF , then E a( ) is a function of this random variable and as a result is a random variable itself. A. CDF of minimum EPG 319 | P a g e
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In order to derive the CDF of the minimum EPG, , we make a change of variable in (7) by introducing , then the inequality
is
equivalent to
where K is called the Rician K-factor and is the power (strength) of the constant line-of-sight (LOS) component relative to the random non-LOS component
where we have used (9) in the appendix to express as a function of t. Due to the positivity of EPG (for nonzero ), we can assume is an event of zero probability) and restrict our attention to . The equivalent inequality
By applying the inverse of the Lambert W function on bothsides yields,
Thus the CDF of the minimum EPG can be expressed as afunction of the fading distribution.
B. Outage EPG as a performance measure When the fading power follows a probability distribution, there may be a range of channels that are very poor in quality or cost too much EPG. In these cases, it may make sense to turn off the radio completely (outage event) until the channel is of a better quality. Service requirements specify the maximum tolerated outage probability po. A percentile is the value of a variable below which a certain percent of observations fall. Thus a (1 − po) ×100% percentile EPG or equivalently po × 100% outage EPG is the highest cost (in EPG) that a user will incur in order to communicate a bit. In a similar manner, outage spectral efficiency gives the slowest rate at which a user will communicate. Outage transmit power gives the maximum power that will be used. In the next section, various antenna configurations will be compared in terms of the outage EPG achieved. The most energy-efficient configuration is defined as that configuration which yields the least outage EPG, given a certain specified po.
V.
NUMERICAL EXAMPLE
The parameters (unless specified otherwise) used in the numerical example are given in table I. EPG is measured in dBmJ which is the energy in dB relative to 1 mJ. We assume that wavelength is calculated as = 3×108fc. A realization of the channel gain is generated by finding the maximum eigenvalue of the Wishart matrix H†H. Each (i, j)-th element in a MIMO channel matrix realization is an i.i.d. random variable with the following distribution www.ijera.com
A. CDF of EPG for various configurations In order to validate the theoretical expression for the CDF of the minimum EPG, we calculated the CDF using simulation by generating 80, 000 channel realizations for each configuration. Figure 2 shows a zoomed-in view of the EPG CDF corresponding to a distance of 25 meters. In Fig. 2, we consider MIMO configurations of 1 × 1, 1 × 2 and 2 × 2 and both Rayleigh fading and Rician fading with K = 10. It can be seen that the theoretical CDF values (solid and dashed lines) match the simulation values (symbols) for all configurations and channel fading conditions considered. In order to evaluate the EPG CDF, we have utilized recent results on the distribution of maximum eigenvalue for non central Wishart matrices. For the Rayleigh fading case, it can be observed that the 99th percentile EPG for the 2×2 case is around −28.2 dBmJ, where as it is around −23.7 dBmJ for the 1×1 configuration. Thus for the Rayleigh fading case, the enhanced reliability (diversity) provided by multiple antennas is vital to providing energy-efficient communication. For the Rician case, we observe that the CDFs, in general,
show less spread due to the constant line-of-sight component and thus lower values of EPG than the Rayleigh fading case. For the Rician case, we also observe that the smaller configurations possess a lower lower 99th percentile EPG than the 2 × 2 configuration. This is because the larger configuration has a higher circuit power consumption. B. Most energy-efficient MIMO configuration The most energy-efficient MIMO configuration or optimum MIMO configuration, 320 | P a g e
A. R. Mekala et al. Int. Journal of Engineering Research and Application Vol. 3, Issue 5, Sep-Oct 2013, pp.317-323 nT,EE × nR,EE NMIMO, is defined to be the MIMO configuration that yields the smallest outage EPG (at a specific po). The optimum configuration is chosen from a discrete and (in practice) finite set of allowed MIMO configurations NMIMO. Since we use a closed-form formula to evaluate the outage EPG, evaluating the optimum MIMO configuration does not require time consuming simulations. We consider balanced MIMO configurations limited to a maximum of four antennas at either
Fig. 2. Cumulative Distribution Function of Energy PerGoodbit at a distance of 40 m. Solid lines correspond to Rayleigh fading. Dashed lines correspond to Rician fading (K = 10). end. For a given total number of antennas, balanced MIMO configurations provide superior diversity performance.We consider balanced MIMO configurations limited to a maximum of four antennas at either end. For a given total number of antennas, balanced MIMO configurations provide superior diversity performance. Since we have assumed symmetric circuit energy costs at both transmitter and receiver, an X × Y configuration is equivalent to a Y × X configuration. It should be noted that the equivalence will generally not hold when there is spatial correlation. In our case, the search space can be reduced by removing equivalent configurations. Thus the set of allowed MIMO configurations are NMIMO = {1 × 1, 1 × 2, 2 × 2, 2 × 3, 3 × 3, 3 × 4, 4 × 4}. Notice that the number of antennas, NA = nT + nR uniquely determines the configuration. If denotes the optimum or most energy-efficient number of antennas,then . Figure 3 shows the optimum MIMO configuration at a 1% outage level (po = 0.01) over a range of μd, over Rayleigh (K = 0) and Rician (K = 10) fading. The largest MIMO configuration is preferred (for both fading scenarios) when transmit power cost dominates circuit power cost, i.e., when www.ijera.com
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μd<< 1. In order to meet the outage requirement, diversity helps to reduce the probability that bad channels need to be used for transmission. As gets smaller than the range shown, we have observed that largerMIMO configurations can provide improvements under suitable conditions. On the other hand, when circuit power cost is very large ( around 105 for this example), SISO (1×1) is the preferred configuration. For Rician fading, SISO is optimal for a larger range of μd which indicates that the LOS component provides some amount of reliability. In general, there is an optimum MIMO configuration depending on the value of μd.
Fig. 3.Energy-efficient MIMO configuration N∗A= nT,EE +nR,EEfor both Rayleigh (K = 0, solid) and Rician (K = 10, dashed) fading. Figure 4 shows the corresponding 1%-outage EPG (equivalently 99th percentile EPG) for the optimum MIMO configuration, the largest MIMO configuration (4 × 4) and the SISO configuration under Rayleigh fading. The Rician fading curves are similar but exhibit lower values of outage EPG and is not shown here. It can be seen that the outage EPGfor the SISO configuration rapidly increases as the transmit power cost increases. Although not as dramatic, the largest MIMO configuration is seen to be suboptimal when circuit power costs dominate. Figure 5 shows the 1%-outage spectral efficiency achievedby the the optimum MIMO configuration, the largest MIMO configuration (4×4) and the SISO configuration (1×1) under Rayleigh fading. Also shown is the average spectral efficiency for the optimum MIMO configuration.We can observe that larger MIMO
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A. R. Mekala et al. Int. Journal of Engineering Research and Application Vol. 3, Issue 5, Sep-Oct 2013, pp.317-323
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applied recent statistical results on the maximum eigenvalue of Wishart matrices to evaluate the most energy-efficient MIMO-MRC configuration. Depending on the ratio of transmit energy to circuit energy, it was observed that there is an optimum number of antennas (or diversity order) that provides the most energy-efficient operation. Although the presence of a LOS component lessens the need for diversity (more antennas) compared to NLOS operation, it was observed that as transmit energy becomes more dominant compared to circuit energy, larger MIMO configurations provided better energyefficient operation in both Rician and Rayleigh fading scenarios. Larger MIMO configurations also provided higher and steadier spectral efficiencies compared to SISO configuration. Fig. 4. Outage EPG (Rayleigh fading) for the Optimum, 4 × 4 and 1 × 1 configurations as a function of the ratio of circuit and transmit EPG for a unit total power and rate, μd= kc/kt. configurations provide larger outage spectral efficiencies, even though these larger configurations are not utilizing multiplexing gain. The jagged nature of the spectral efficiency for the optimum MIMO configuration is due to the optimum number of antennas changing as a function of μd. Ignoring the jaggedness, we can observe that the spectral efficiency increases as the circuit power cost proportion increases and this makes sense because circuit energy isminimized by minimizing transmission time or maximizing spectral efficiency.
Fig.5.Optimum, 4×4 and 1×1 configurations,Outage spectral efficiency (Rayleigh fading)
VI.
CONCLUSION
An energy minimization problem was formulated by considering an objective function that corresponded to the energyper-goodbit (EPG) of a wireless system employing a MIMO-MRC scheme.A closed-form solution of the energy-efficient power allocation was derived. A closed-form expression for theminimum EPG was also obtained and was used to derivethe CDF of the minimum EPG as a function of the CDF of the fading distribution. A numerical example www.ijera.com
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